SELECT * FROM question_mgmt as q WHERE id=5147 AND status=1 SELECT id,question_no,question,chapter FROM question_mgmt as q WHERE courseId=9 AND subId=6 AND chapterId=268 and ex_no='1' AND status=1 ORDER BY CAST(question_no AS UNSIGNED) CBSE Free NCERT Solution of 10th mathematics Real Numbers use euclid rsquo s division lemma to show that the

Question:

Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.
[Hint : Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square each of these and show that they can be rewritten in the form 3m or 3m + 1.]

Answer:

Let x be any positive integer and b = 3.

Then, by euclid’s algorithm

x = 3q + r, where q ≥ 0 and r = 0, 1, 2          [0 ≤ r ≤ b]

Case (i) :    For r = 0, x = 3q, = x2 = 9q2, taking 3 as common,

                   x2 = 9q2  = 3 (3q2), which is of the form 3m, where m = 3q2.

Case (ii) :   For r = 1, x = 3q + 1

                   x2 = 9q2 + 1 + 6q, taking 3 as common,

                   = 3 (3q2 + 2q) + 1, which is of the form 3m + 1, where m = 3q2 + 2q

Case (iii) :   For r = 2, 3q + 2

                   x2 = 9q2 + 4 + 12q = (9q2 + 12q + 3) + 1, taking 3 as common,

                   = 3 (3q2 + 4q + 1) + 1, which is of the form 3m +1, where m = 3q2 + 4q + 1

                   Hence, x2 is either of the form 3m, 3m + 1 for some integer m.


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